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A195484 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).

%I A195484 #11 Jan 27 2018 02:29:03
%S A195484 1,7,0,6,0,4,6,3,5,0,3,4,4,2,3,2,4,4,2,2,8,5,4,1,9,9,0,4,0,9,8,4,7,0,
%T A195484 6,0,7,6,2,3,6,8,0,2,8,8,7,3,0,0,1,5,3,3,5,0,3,6,2,4,1,9,6,8,3,9,0,7,
%U A195484 0,1,0,6,1,2,2,0,0,2,7,4,7,9,4,9,7,7,8,4,3,2,5,8,8,0,1,6,8,6,3,5
%N A195484 Decimal expansion of shortest length, (B), of segment from side BC through centroid to side BA in right triangle ABC with sidelengths (a,b,c)=(sqrt(2),sqrt(5),sqrt(7)).
%C A195484 See A195304 for definitions and a general discussion.
%H A195484 G. C. Greubel, <a href="/A195484/b195484.txt">Table of n, a(n) for n = 1..10000</a>
%e A195484 (B)=1.7060463503442324422854199040984706076236802887300...
%t A195484 a = Sqrt[2]; b = Sqrt[5]; h = 2 a/3; k = b/3;
%t A195484 f[t_] := (t - a)^2 + ((t - a)^2) ((a*k - b*t)/(a*h - a*t))^2
%t A195484 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195484 f1 = (f[t])^(1/2) /. Part[s, 4]
%t A195484 RealDigits[%, 10, 100] (* (A) A195483 *)
%t A195484 f[t_] := (t - a)^2 + ((t - a)^2) (k/(h - t))^2
%t A195484 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195484 f2 = (f[t])^(1/2) /. Part[s, 4]
%t A195484 RealDigits[%, 10, 100] (* (B) A195484 *)
%t A195484 f[t_] := (b*t/a)^2 + ((b*t/a)^2) ((a*h - a*t)/(b*t - a*k))^2
%t A195484 s = NSolve[D[f[t], t] == 0, t, 150]
%t A195484 f3 = (f[t])^(1/2) /. Part[s, 1]
%t A195484 RealDigits[%, 10, 100] (* (C) A195485 *)
%t A195484 c = Sqrt[a^2 + b^2]; (f1 + f2 + f3)/(a + b + c)
%t A195484 RealDigits[%, 10, 100] (* Philo(ABC,G) A195486 *)
%Y A195484 Cf. A195304.
%K A195484 nonn,cons
%O A195484 1,2
%A A195484 _Clark Kimberling_, Sep 19 2011